The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global wellposedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension d ≥ 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d ≥ 5, and a careful study of the nonlinear structure of the quadratic terms in dimension 3 and 4 involving the theory of space time resonance. RésuméLeséquations d'Euler-Korteweg sont une modification deséquations d'Euler prenant en compte l'effet de la capillarité. Dans le cas général elles forment un système quasi-linéaire qui peut se reformuler comme uneéquation de Schrödinger dégénérée. L'existence locale de solutions fortes aété obtenue par Benzoni-Danchin-Descombes en toute dimension, mais sauf cas très particuliers il n'existe pas de résultat d'existence globale. En dimension au moins 3, et sous une condition naturelle de stabilité sur la pression on prouve que pour toute donnée initiale irrotationnelle petite, la solution est globale. La preuve s'appuie sur une estimation d'énergie modifiée. En dimension au moins 5 les propriétés standard de dispersion suffisent pour conclure tandis que les dimensions 3 et 4 requièrent uneétude précise de la structure des nonlinéarités quadratiques pour utiliser la méthode des résonances temps espaces.

While the non-homogeneous boundary value problem for elliptic, hyperbolic and parabolic equations is relatively well understood, there are still few results for general dispersive equations. We define here a convenient class of equations comprising the Schrödinger equation, the Airy equation and linear "Boussinesq type" systems, which is in some sense a generalization of strictly hyperbolic equations, and for which we define a generalized Kreiss-Lopatinskiȋ condition. From the construction of generalized Kreiss symmetrizers (adapted from hyperbolic theory) we deduce a priori estimates and well posedness for the pure boundary value problems (BVP) on a half-space associated to this class of equations. The initial boundary value problem (IBVP) is investigated too for the special case of the Schrödinger equation, and possible generalizations of the proof for other problems are indicated.

In this paper we study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When Ω is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If Ω c is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.